Google Sites now provides Google Drive Video embedding. This means you can now make use of some advantages over YouTube video embedding: add videos stored in Google Drive that do not provide links to related videos; and you can share the videos privately with more than 25 users, e.g. a whole Google Apps domain. The embed tool also lets you pick a custom size for the video without the need to use custom code.
You can already embed videos from YouTube in your Google Site so why use Google Drive Videos? If you do not want to display links to related videos, as YouTube does by default, Google Drive videos do not provide these links (you can disable these links on YouTube videos but you then need to use custom code to embed them in your site). YouTube only allows you to share a video with 25 people: Google Drive videos does not impose any limit and if you use Google Apps you can take advantage its sharing options.
The full-screen button does not work at the moment. Google know of this issue and are working on a fix.
None: Google have fixed the full-screen button so it works correctly now.
You can get this feature now on stand-alone/consumer/personal Google Sites and Google Apps domains with the Rapid Release track enabled.